Abstract :
Let image be a finite dimensional complex simple Lie algebra and U(image) its enveloping algebra. The quantum group of Drinfeld and Jimbo is a Hopf algebra denoted Uq(image) defined on Chevalley-like generators over image[q, q−1]. Through "specialization" of q at different ε set membership, variant image one obtains a parameterized family of Hopf algebras. When ε = 1 one recovers the classical universal enveloping algebra. Moreover, when ε is not a root of unity Lusztig (Adv. Math.70 (1988), 237-249) and Rosso, (Comm. Math. Phys. 117 (1988), 581-593) have shown independently that the representation theory of Uε(= Uε(image)) is analogous to that of U(image). When we fix ε a primitive root of unity the situation changes considerably. At odd roots of unity the representations of Uε are partitioned by conjugacy classes of the algebraic group G with Lie algebra image [DC-K]. This paper relates the representations of Uε at even roots of unity to conjugacy classes in the group Glogical or-the Langlands dual of G. The partition is somewhat finer than that at odd roots of unity and requires a more detailed analysis. This correspondence is then used to study the representations at even roots of unity. In particular, we obtain a "triangulability" result which allows us to calculate the degree of Uε using a deformation argument.
